Question

3. Find the horizontal, vertical, or oblique asymptote(s) for the following functions. If there is no asymptote, say so. 9 (a) y=r- (b) y= (C) y = 1-12 (d) y = 22 +4 12+1

Answer 1

Important 17 vertical Asymptotes of set of XER sot denominator of fex) must be zero. 2) Horizontal Asymptoteso- i) If degree of numerator < degree of denominator then: yao i Horizontal Asymptose ") If degrere of numerator = degree of denomi- -hater then leading coefficient of numerater leading coofficient of denominator ili) degree of numerator 7 degree of denominaton then & Horizontal asymptote ya

solution y = a - 9 x²g - y = x for 2-0 dnominator goes to o "dzo is vertical asymptote. & degree of cx² g) > deg Cx) It has no Horizontal asymptore Now x -9 2 & 2 y = x is oblique asymptote y = 22 1-x2 x=+1 gives us denominater o 821 & X=1 are Versical asymptore Now degree of c1) = 0 & degree of cl-x²) = 2 4 ol2 - yoo in Horizonal asymptore degree of neumerater o less than denonai nater deare ) it has no oblique asymptote.

y = 22 4- x² 2 goes to o for x= x= -2 denominator ir 8=2 f x=-2 are vertical aymptote. degree of numerator degree of denominator it has no horizontal asymptote. Now x² - 42 y = -x is oblique asymptote @ y = x²+1 x2 + 4 XER site x²1420 hence it has no vertical asymptote degree (x²) 1) = degree (x²+4) کی ملا 2) = 21 1 y = 1 is Horizontal asymptote degree cx²4 ) = degree (2²+4) z) it has no oblique asymptote. Important fox) has oblique asymptote iftits Numerator has excitly cleare more than one degree of denominator. -fy quotient is oblique asymptote.